Simulation of four-bar linkages
With SVG and Flash embedded graphics

Introduction

The four-bar linkage is a versatile planar mechanism, used extensively in articulated systems:

In wheel vehicles: For the Ackerman steering in all automobiles and trucks. For automotive and bicycle suspensions. For hydraulic tail lifts, and scissor tippers on dump trucks. In hydraulic machinery: For tilting the bucket on hydraulic excavators and loaders. For the blade actuators of earth moving equipment. For the three-point hitch of agricultural tractors. In industrial automation and material handling: For the needle mechanism on sewing machines. For scissor lifts and conveyor ancillary equipment. For packaging machines. In doors and covers, instead of a simple hinge: For garage lifting doors. For engine hoods. In hand tools: For locking pliers. For bolt cutters. For manual clamps. Etc.

Figure 2: a four bar linkage and his coupler curve

Empirical design

In mechanical engineering, when you are designing a four bar mechanism for a given application, there is no analytic way for computing the parameters of such a mechanism. Consequently, different techniques have been developed for finding the best parameters with empirical methods:

• In order to find a path for your application, you can refer to the standard Hrones and Nelson Atlas of four bar mechanisms:
  in 1951, these two scientists established at the MIT a large catalogue of coupler curves, generated with a mechanical simulator.
• If you want to experiment your own parameters, you can also compute and simulate the geometry on a computer,
  and use vector graphics for displaying the results of the 2D simulation.

Figure 3: a coupler curves from the Hrones and Nelson Atlas

Parts of a four-bar linkage

Figure 4: parts and parameters of a four-bar linkage.

The mechanism of a four-bar linkage is composed of:

• Four mechanical parts:
• A fixed link OD (considered as the static frame of the machine).
• A crank lever OA (considered as a pilot element).
• A connecting rod AB.
• A rocker lever DB (considered as a follower element).
• Four simple hinges for the joints O A B D between consecutive links.
• A coupler point C fixed on the connecting rod:
• When A, B and C are aligned, the rod is a straight lever.
• When A, B and C are not aligned, the rod has a triangular shape.

Parameters of the mechanism

The geometry of the mechanism is defined by the following parameters:

• 6 dimensional parameters:
• The length of each bar:
• lp for the crank lever.
• lq for the connecting rod.
• lr for the rocker lever.
• ls for the static frame.
• The coordinates of the coupler point, which may be specified by:
• Cartesian coordinates cu and cv, relative to the base segment AB.
• Or polar coordinates, sometimes used in other simulators.
• 3 variable parameters, computed by the simulation loop:
• The rotation angle φ of the crank.
• The rotation angle ψ of the rocker.
• The rotation angle θ of the connecting rod.
• Different utility parameters:
• Computed length and angles of the segment AD
• Computed position of the coupler point C

The following table summarizes these parameters:

Geometric analysis

Computation of the position with trigonometry

To compute the position of the joint A on the crank, we use the input angle φ :

xA = xO + lp * cos( φ )
yA = yO + lp * sin( φ )

To compute the position of the joint B between connecting rod AB and the rocker BD :

• The rotation angle of the rocker BD is double-valued and
  it depends on whether the point B is below or above the segment AD.
• On the geometry, it means that joint B is on the intersections
  of two circles having a radius of lq and lr, centered on the joints A and B.
• To compute the position of B, we use the the 3 triangles ABD, AHB, and DHB.
• To compute these triangles, we study the right triangle DHB.

To compute the β angle in the triangle DHB :

1. The variable lt = distance( A, D ) is defined by:
   lt = sqrt( (xD - xA)² + (yD - yA)²)
2. The lt length is the sum of AH and HD :
   lt = AH + HD
3. On the right triangle AHB, the Pythagorean theorem defines the expression :
   AB² = AH² + HB²
4. On the right triangle DHB, the Pythagorean theorem defines the expression :
   DB² = DH² + HB²
5. The subtraction of the 3^rd and 4^th equations gives :
   DA² - DB² = lq² - lr² = AH² - HB²
6. The 2^nd and 5 ^th equations can be combined :
   lq² - lr² = (AH + HB) * (AH - HB) = lt * (AH – HB)= lt * (lt - 2 * HB)
7. The distance HB is defined by the 6^th equation :
   HB = (lt² - lq² + lr²) / (2 * lt)
8. The distance HB and the radius lr define the cosine of the β angle:
   cos( β )= HR / lr
9. The 7^th and the 8^th equations give :
   β = acos( (lt² - lq² + lr²) / (2 * lr * lt) )

To compute the α angle between the segment DA and the static bar OD :

α = atan2( yA – yD, xA - xD )

Now, we can compute the two possible ψ angles between the rocker DB and the static bar OD :

ψ = α ± β

The sign of the β angle is:

• <0 when the joint B is above the line segment AD (case of the direct circuit configuration).
• >0 when the joint B is below the line segment AD (case of the crossed circuit configuration).

Now, we can compute the two positions of the B joint :

xB = lr * cos( ψ ) + ls
yB = lr * sin( ψ )

Now, we can compute the θ angle (-π ≤ θ ≤ π ) between the rod AB and the static bar OD :

θ = atan2( yB – yA, xB - xA )

Finally, we can compute the position of the coupler point C, using the relative coordinates cu and cv :

xC = xA + cu * cos( θ ) - cv * sin( θ )
yC = yA + cu * sin( θ ) + cv * cos( θ )

Usage of the Grashof condition

S = shortest link, L = longest link	Source: Wikipedia
Figure 5: types of four bar linkages

The Grashof condition determines different types of four-bar linkages:

• The mechanism can have either a continuous or an alternative rotation:
• Depending of the dimension of each link, there are different kinds of four-bar linkages,
  the crank and the rocker may either follow a complete rotation, or just oscillate on a limited dwell angle.
• When designing a linkage where the input linkage is continuously rotated (for example, driven by a motor),
  it is important that the input link can freely rotate through complete revolutions.
• The Grashof's law provides a simple test for checking this condition:
• For a planar four bar linkage having a continuous relative rotation between two members,
  the sum of the shortest and longest links must be less than the sum of the remaining links.
• In other words, if the crank has to rotate on a whole turn, the mechanism must respect the following conditions:
• S + L < P + Q
• S and L are the lengths of the shortest and the longest links respectively.
• P and Q are the lengths of the remaining two links.
• The figure above shows different configurations:
• The Grashof condition is respected in the 1^st, 2^nd and ^4th cases.
• For a drag link configuration, S is the length of the fixed link ls.
• For a crank-rocker configuration, S is the length of the crank lp.
• For a double-rocker configuration, the linkage will lock at the ends of the stroke of the crank
• For a parallelogram configuration, the mechanism has 2 mathematic singularities,
  but the two cranks can still follow a whole turn because we have S = P  and L = Q.

The four-bar simulation software

Usage of vector graphics for displaying the simulation sequence

Here we use SVG and Flash techniques for displaying the simulation:

• SVG for generating the analysis report in XML format.
• Flash for embedding it in a Windows application.

Configuration files and result files

• Configuration files:
• The simulator is configured with the text files in the test/config directory.
• In order to load a configuration file, you can:
• Either run the program with the path of a configuration file:
  four_bar.exe -f config/crank_rocker.txt
• Or just drag another configuration file over the window of the simulator:
  the mechanism is rebuilt as specified in the configuration file.
• Result files:
• For each simulation cycle, the program generates .SVG and .SWF files.
• If you want to save them, you can rename or copy the last .SVG or .SWF files generated in the test directory.
• The Flash movie of a simulation can be embedded in a HTML page with the following syntax:
  

  <embed src="triangle_path.swf" width="400" height="300" quality="high" type="application/x-shockwave-flash" >

  Figure 6: sample embedded flash movie

Demonstration with well known mechanisms

Straight line linkages

In the steam engines developed in the 19^th century, the translation of the piston is often guided by an articulated system, instead of a sliding system. For such applications, different inventors, engineers and mathematicians disvovered interesting four bar linkages where a part of the coupler curve is almost a straight line.

These mechanisms only approximate a straight line. An exact straight line linkage was discovered in 1864 and 1871 by Peaucellier and Lipkin:

• Charles Nicolas Peaucellier (1832-1913), French military engineer.
• Independently discovered by Lipmann I. Lipkin (1846-1876), Lithuanian mathematician.
• Although it is mathematically perfect, it is not presented here because it is complex (8 independent solids)
  whereas the straight line linkages presented above are still used today in various applications.

References

1. Introduction to the four bar linkages:
   See on Wikipedia – http://en.wikipedia.org/wiki/Four_bar_linkage
2. History of the straight line linkages:
   "How to Draw a Straight Line", by Daina Taimina – http://kmoddl.library.cornell.edu/tutorials/04/
3. Analysis of the Four-Bar Linkage
   by John A Hrones and George L. Nelson – M.I.T. Press and John Wiley & Sons – 1951
4. A animated presentation of the Hrones and Nelson coupler curves:
   Matlab simulation developed by Fan (Michael) Mo
   taken from the page 116 of "Design of Machinery", 3^rd edition – by Robert L. Norton –  McGraw-Hill – 2004
   http://dynaden.wonkwang.ac.kr/data file/kinematics/animation/WM/ch3/coupler_curve_atlas/coupler_curve_atlas.htm
5. Animated images showing straight line linkages:
   at the IEL department at UC Davis – http://iel.ucdavis.edu/design/fourbar/fourbarStraight.html
6. Synthesis of Four-Bar Linkage Mechanisms using Pattern-Matching Approach and Genetic Algorithms
   by Vipul Kumar and Gupta Shorya Awtar – http://www.cse.iitk.ac.in/~amit/courses/751/97/LINKAGE_SYNTHESIS 
7. An open-source simulator of four-bar linkages :
   "Logiciel de visualisation de mécanismes à quatre barres" – 1995
   by Jean-Pierre Merlet – INRIA Sophia Antipolis – ftp://ftp-sop.inria.fr/coprin/4bar/
6. A commercial simulator of four-bar linkages :
   Four-Bar Linkage Analysis and Synthesis – http://www.softintegration.com/chhtml/toolkit/mechanism/fourbar/
7. A commercial simulator for designing different types of linkages :
   Linkage Design Tools – by Robert L. Norton – http://www.designofmachinery.com/Linkage/index.html